Exploring the Possibilities of a New Relativistic Quantum Theory

[A. Neumaier]

It is possible to make a source of light, which produces single photons one-by-one on demand. If I shine this light on a double-slit, I get a famous picture on the screen, where each photon makes a separate tiny spot, and only after long exposure the interference pattern emerges. I don't see other way to explain this behavior but the quantum mechanical picture in which light particles are described by a probabilistic wave function. The Maxwellian field representation of light is incapable of describing this experiment at all.

See
http://arnold-neumaier.at/ms/lightslides.pdf
http://arnold-neumaier.at/ms/optslides.pdf
for the standard quantum optics view of this matter. It is all explainable by the field picture - particles entering only in a semiclassical view.

 

[meopemuk]

See
http://arnold-neumaier.at/ms/lightslides.pdf
http://arnold-neumaier.at/ms/optslides.pdf
for the standard quantum optics view of this matter. It is all explainable by the field picture - particles entering only in a semiclassical view.

So, photons are "localized lumps of energy". Do they pass through one slit or through both slits at once in the double-slit experiment?

Eugene.

 

[A. Neumaier]

So, photons are "localized lumps of energy". Do they pass through one slit or through both slits at once in the double-slit experiment?

As any wave, localized or not, through both slits.

Only upon recording the photons, they materialize - at a single spot only.

 

[A. Neumaier]

Suppose now that we constructed a relativistic interacting theory in which the interacting Hamiltonian is *not* a product of fields. For example, it can be $$V= a^{\dag}a^{\dag}aa$$.

This still is a product of nonrelativistic local fields. I guess you mean: not (a linear combination of) products of relativistic, local fields.

Then [...] two important conditions of Haag's theorem will not be satisfied, and we will not be able to prove that the Fock space is excluded.

Yes.

As a result of this exercise we will obtain a non-trivial interacting theory in the Fock space. "Dressed particle" theories are exactly of this form. Their only problem is that interacting fields are non-covariant and non-commuting. Could you please explain why you think that this is an important problem? Is there any measurable property that proves the impossibility of non-covariant and non-commuting interacting fields?

First, you now have infinitely many interacting terms. So additional problems about well-definedness and self-adjointness arise.

Second, observed physics is Poincare invariant, to very high accuracy. Your Poincare representation is only approximate at any order.

Third, lack of covariance makes all calculations (especially those of higher order) much more messy. This is the main reason, I think, why very few people today work with noncovariant methods.

Finally, you do everything at zero temperature. However, most application of QED where the time evolution is important take place at finite temperature. There everything is different again - in place of the asymptotic particles at T=0 one now has only effective particles, which are different.

 

[meopemuk]

As any wave, localized or not, through both slits.

Only upon recording the photons, they materialize - at a single spot only.

For the double-slit experiment with visible light the distance between two slits can be macroscopic, e.g. 0.1 millimeter, or something like that. This means that the photon "lump" should be no smaller than this size. So, you are saying that the energy lump associated with a single visible-light photon can be as big as 0.1 millimeter or so? And that all this volume is filled with a time-changing electromagnetic field? Do I understand your model correcly?

Eugene.

 

[A. Neumaier]

For the double-slit experiment with visible light the distance between two slits can be macroscopic, e.g. 0.1 millimeter, or something like that. This means that the photon "lump" should be no smaller than this size. So, you are saying that the energy lump associated with a single visible-light photon can be as big as 0.1 millimeter or so? And that all this volume is filled with a time-changing electromagnetic field?

Of course. Photons can be very delocalized. This is precisely what happens in a double slit experiment. Each photon in a laser beam has the same shape as the classical field by which this beam is described.

My views about what photons are turned by almost 180 degrees after I had begun to talk to the experimentalists in Zeilinger's group (who moved to Vienna in 1999)...

 

[meopemuk]

This still is a product of nonrelativistic local fields. I guess you mean: not (a linear combination of) products of relativistic, local fields.

In our example we've considered a $$\Phi^4$$ theory. So, I am saying that $$V = a^{\dag} a^{\dag}aa$$ cannot be expressed as a linear combination of integrals

$$\int d\mathbf{r} \Phi^n(\mathbf{r}, 0)$$

First, you now have infinitely many interacting terms. So additional problems about well-definedness and self-adjointness arise.

What do you mean by "many interaction terms"? My interaction has only one term $$V = a^{\dag} a^{\dag}aa$$.

Second, observed physics is Poincare invariant, to very high accuracy. Your Poincare representation is only approximate at any order.

So, you are saying that it is impossible to formulate a Poincare-invariant theory with interaction $$V = a^{\dag} a^{\dag}aa$$? Can you prove that?

You are making the claim that Haag's theorem does not leave any chance for using Fock space in relativistic quantum theories. So, it is your job to prove that interaction $$V = a^{\dag} a^{\dag}aa$$ violates relativity.

Third, lack of covariance makes all calculations (especially those of higher order) much more messy. This is the main reason, I think, why very few people today work with noncovariant methods.

Messy or not messy is a matter of taste and convenience. This is not a scientific argument.

Finally, you do everything at zero temperature. However, most application of QED where the time evolution is important take place at finite temperature. There everything is different again - in place of the asymptotic particles at T=0 one now has only effective particles, which are different.

I am interested in what happens when two electrons (or other particles) move close to each other. I think it is important to understand this simplest event first. Only then we can switch to systems with many particles, non-zero temperature, etc.

Eugene.

 

[A. Neumaier]

What do you mean by "many interaction terms"? My interaction has only one term $$V = a^{\dag} a^{\dag}aa$$.

So you are talking about a harmonic oscillator?

Or do you mean $$V = \int dx a^*(x) a^*(x)a(x)a(x)$$? This is already a linear combination of field products, not only one product.

So, you are saying that it is impossible to formulate a Poincare-invariant theory with interaction $$V = a^{\dag} a^{\dag}aa$$? Can you prove that?

No. What I was saying is spelled out in post #74.

What you though I was saying is probably possible in the style of Bakamjian, at least when one allows an additional energy shift. But then you don't have cluster separability.

You are making the claim that Haag's theorem does not leave any chance for using Fock space in relativistic quantum theories. So, it is your job to prove that interaction $$V = a^{\dag} a^{\dag}aa$$ violates relativity.

No. I am only claiming that Haag's theorem says what it says. This is in the literature, freeing me from any further obligation.

If you create a theory that violates the assumptions of Haag's theorem, you are not bound to its conclusions.

Messy or not messy is a matter of taste and convenience. This is not a scientific argument.

Of course it is. Kepler's theory was in his time not more accurate than Ptolemy's. But it was far less messy, and made things so much easier that it was adopted as the standard.

I am interested in what happens when two electrons (or other particles) move close to each other.

Whereas I an interested to explain standard QFT to those who want to understand it.

I think it is important to understand this simplest event first. Only then we can switch to systems with many particles, non-zero temperature, etc.

2-electron systems are useless approximations, once one has interactions terms changing particle number.

Once one wants to insist on a dynamical view (rather than the asymptotic scattering view), one needs to prepare the particles at finite time, hence they'll never be exact 2-particle states (which make sense only asymptotically). But even if they were exact for one moment, they'd lose that property the very next moment.

Thus you are chasing a chimera. I prefer to chase the insights of the community of high energy physicists, who all work in the quantum field paradigm.

 

[meopemuk]

So you are talking about a harmonic oscillator?

Or do you mean $$V = \int dx a^*(x) a^*(x)a(x)a(x)$$?

No, I meant a 2-particle interaction whose full expression involves momentum-space a/c operators

$$V = \int d\mathbf{p} d\mathbf{q} d\mathbf{k} V(\mathbf{p}, \mathbf{q}, \mathbf{k}) a^{\dag}(\mathbf{p-k}) a^{\dag}(\mathbf{q+k}) a(\mathbf{p}) a(\mathbf{q})$$

Here $$V(\mathbf{p}, \mathbf{q}, \mathbf{k})$$ is a numerical coefficient function, which can be selected in such a way that interaction is translationally and rotationally invariant. I also claim that one can find a corresponding interacting boost operator, so that entire theory becomes relativistically invariant.

My other claim is that in this theory interacting quantum fields constructed by usual formulas do *not* transform covariantly. So, the main condition of Haag's theorem is not satisfied, the theorem is not applicable, and we are permitted to work in the Fock space.
More details on this example can be found in

H. Kita, "A non-trivial example of a relativistic quantum theory of particles without divergence difficulties", Progr. Theor. Phys., 35 (1966), 934.

If you create a theory that violates the assumptions of Haag's theorem, you are not bound to its conclusions.

Great! I am glad that we agree about that.

Eugene.

 

[meopemuk]

Of course. Photons can be very delocalized. This is precisely what happens in a double slit experiment. Each photon in a laser beam has the same shape as the classical field by which this beam is described.

OK, so we have a macroscopic lump of electromagnetic energy, which falls on the double slit and interferes with itself according to Maxwell equations. Then this macroscopic lump reaches the photographic plate and suddenly collapses to a microscopic point, whose size is comparable to the size of a grain of photo-emulsion. The photon energy, that was previously spread up in a macroscopic lump now gets released within a group of few atoms.

How does Maxwell equation explain this collapse?

Eugene.

 

[A. Neumaier]

No, I meant
$$V = \int d\mathbf{p} d\mathbf{q} d\mathbf{k} V(\mathbf{p}, \mathbf{q}, \mathbf{k}) a^{\dag}(\mathbf{p-k}) a^{\dag}(\mathbf{q+k}) a(\mathbf{p}) a(\mathbf{q})$$

OK, not a product but an integral over a product. It is difficult to understand you if you shorten formulas to an extent that they become unrecognizable.

I also claim that one can find a corresponding interacting boost operator, so that entire theory becomes relativistically invariant.

My other claim is that in this theory interacting quantum fields constructed by usual formulas do *not* transform covariantly.

The quantum fields of interest are those constructed from
$$\phi(0)=\int dp (a(p) +a^*(p))$$
by conjugating with the representation of the Poincare group you claim exists. Since the stabilizer of zero is the Lorentz group, the translations gives a Hermitian field phi(x) which transforms covariantly. But the field is not local, and hence not of interest for particle physics. One can construct plenty of similar field with Bakamjian's construction.

So, the main condition of Haag's theorem is not satisfied, the theorem is not applicable, and we are permitted to work in the Fock space.

Yes, but you get something lacking cluster separation.

More details on this example can be found in

H. Kita, "A non-trivial example of a relativistic quantum theory of particles without divergence difficulties", Progr. Theor. Phys., 35 (1966), 934.

Probably this is the reason why this pseudo-breakthrough only generated 7 references in over 50 years.

 

[A. Neumaier]

OK, so we have a macroscopic lump of electromagnetic energy, which falls on the double slit and interferes with itself according to Maxwell equations. Then this macroscopic lump reaches the photographic plate and suddenly collapses to a microscopic point, whose size is comparable to the size of a grain of photo-emulsion. The photon energy, that was previously spread up in a macroscopic lump now gets released within a group of few atoms.

How does Maxwell equation explain this collapse?

The Maxwell equations are valid in vacuum and need not explain their failure in the presence of a detector.

The behavior of the detector in the presence of the incident classical electromagnetic field is fully explained by the detector's quantum structure. See Chapter 9 of the quantum optics book by Mandel & Wolf.

That the photon ''suddenly collapses to a microscopic point, whose size is comparable to the size of a grain of photo-emulsion'' is pure fantasy. The photon is absorbed by the detector, and doesn't survive as a localized photon.

 

[meopemuk]

The Maxwell equations are valid in vacuum and need not explain their failure in the presence of a detector.

The behavior of the detector in the presence of the incident classical electromagnetic field is fully explained by the detector's quantum structure. See Chapter 9 of the quantum optics book by Mandel & Wolf.

That the photon ''suddenly collapses to a microscopic point, whose size is comparable to the size of a grain of photo-emulsion'' is pure fantasy. The photon is absorbed by the detector, and doesn't survive as a localized photon.

So, is it correct to say that the macroscopically delocalized lump of the photon's EM field is absorbed by the entire photographic plate, and then all this absorbed energy gets channeled somehow to a single grain of photoemulsion?

If this model is not correct, then how should I visualize the interaction between the photon and the photographic plate? Do I need to take the quantum structure of the photographic plate into account?

Eugene.

 

[A. Neumaier]

So, is it correct to say that the macroscopically delocalized lump of the photon's EM field is absorbed by the entire photographic plate

I answered in the thread https://www.physicsforums.com/showthread.php?p=3165407 where this part of the discussion belongs.

 

[meopemuk]

OK, not a product but an integral over a product. It is difficult to understand you if you shorten formulas to an extent that they become unrecognizable.

Sorry about that.

The quantum fields of interest are those constructed from
$$\phi(0)=\int dp (a(p) +a^*(p))$$
by conjugating with the representation of the Poincare group you claim exists. Since the stabilizer of zero is the Lorentz group, the translations gives a Hermitian field phi(x) which transforms covariantly.

You've probably meant

$$\Phi(\mathbf{0}, 0) = \int \frac{d \mathbf{p}}{\sqrt{\omega_{\mathbf{p}}}}\left(a(\mathbf{p}) + a^{\dag}(\mathbf{p}) \right)$$

We went throgh this exercise in posts #135, #137, #139 of the thread
https://www.physicsforums.com/showthread.php?t=388556&page=9 and we found there that covariant transformation law and space-like commutativity can be proven if the interaction is written in the form

$$\int d \mathbf{r} \Phi^n(\mathbf{r}, 0)$$

My interaction does not have this form, so I don't see how you're going to prove the covariance and commutativity?

But the field is not local, and hence not of interest for particle physics.

I don't think there is a proof that particle physics can be explained only in terms of local quantum fields. In my opinion, there is some historically motivated prejudice, but not a proof.

Yes, but you get something lacking cluster separation.

The cluster separability of the "dressed particle" approach is proven in section 10.2. Interaction of the type presented above is cluster-separable if the coefficient function $$V(\mathbf{p}, \mathbf{q}, \mathbf{k})$$ is smooth, i.e., non-singular. This is not difficult to achieve.

Eugene.

 

[A. Neumaier]

You've probably meant
$$\Phi(\mathbf{0}, 0) = \int \frac{d \mathbf{p}}{\sqrt{\omega_{\mathbf{p}}}}\left(a(\mathbf{p}) + a^{\dag}(\mathbf{p}) \right)$$

Well, as I had said in our other discussion, I normalize the annihilator fields differently in order to get rid of the sqrt factors. (Anyway, it doesn't matter in this case, since no matter which p-dependent factor we use, we don't get a causal field.)

My interaction does not have this form, so I don't see how you're going to prove the covariance and commutativity?

This is precisely the point, with my recipe to construct a covariant field you don't get causality. With your recipe to construct a causal field you don't get covariance. But one needs both. Thus either field is useless. And the lack of citations of Kita's paper is empirical proof of that - only 7 people in over 50 years found it useful enough to merit a citation.

The cluster separability of the "dressed particle" approach is proven in section 10.2. Interaction of the type presented above is cluster-separable if the coefficient function $$V(\mathbf{p}, \mathbf{q}, \mathbf{k})$$ is smooth, i.e., non-singular. This is not difficult to achieve.

Please give the page; I didn't see it. I only found a passing reference on p.356, which referred to p.271, which establishes cluster separability only in a preferred frame of reference. You declare this to be enough on p.428 - but very few people will follow you in that. You discard the basic insight of 20th century physics. There are no experimental hints that Nature has preferred frames.

While searching for information, I came across the following unrelated remark on p. 426: ''We can record time even if we do not measure anything, even if there is no physical system to observe.'' I wonder how you record time without observing (i.e., measuring) a clock. If there is no physical system to observe, there is in particular no clock and no observer.

And on p.427; ''the formally quantum nature of clocks and rulers does not play any role in experimental physics.'' But they play a big role if the times and distances become short enough. Accurate experiments become impossible (or extremely difficult) when these are so short that the quantum nature starts playing a role.

 

[meopemuk]

And the lack of citations of Kita's paper is empirical proof of that - only 7 people in over 50 years found it useful enough to merit a citation.

If you want to make fun of Kita's paper, I can help you. One of these 7 citations is in my book, two of them are by my friend Shirokov in Dubna, and three are Kita's self-citations. So, we are not talking about mainstream here. You are right.

Eugene.

 

[meopemuk]

This is precisely the point, with my recipe to construct a covariant field you don't get causality. With your recipe to construct a causal field you don't get covariance. But one needs both. Thus either field is useless.

I agree that the interacting field built in this manner is useless. It is useless not because it is non-covariant or non-commuting, but because actual calculations of the S-matrix, bound states, time evolution, etc. do not involve this field at all. All we need for practical calculations is the Hamiltonian, and there is no need to worry about the properties of the interacting field.

Please give the page; I didn't see it. I only found a passing reference on p.356, which referred to p.271, which establishes cluster separability only in a preferred frame of reference. You declare this to be enough on p.428 - but very few people will follow you in that. You discard the basic insight of 20th century physics. There are no experimental hints that Nature has preferred frames.

I agree that the point about cluster separability is not well emphasized. I will think how to rewrite section 10.2 to make this point more clear. But the idea is very simple. Interaction is guaranteed to be cluster-separable if the coefficient function is smooth. This is Statement 7.7 on page 271. This is the same condition as in Weinberg's sections 4.3-4.4.

In building the "dressed particle" version of QED I start from the usual QED Hamiltonian, whose interaction V satisfies the above separability condition already. (Well, strictly speaking, this condition is satisfied only if we ignore the singularity associated with the zero photon mass. But I don't want to open this can of worms again here.) I obtain the dressed particle interaction by applying an unitary transformation $$e^{i\Phi}$$ to the QED Hamiltonian. As I claim in Theorem 10.2, the Hermitian operator $$\Phi$$ must be smooth (=separable) in order to preserve the S-matrix. So, I pay a special attention to make sure that $$\Phi$$ satisfies this condition. Then, when I do dressing transformation order-by-order in section 10.2, I obtain new interaction terms in the form of multiple commutators involving operators V and $$\Phi$$. Both of these operators are smooth (=separable), so by theorem 7.11 all their commutators are smooth (=separable) too. This proves that the dressed particle Hamiltonian is separable in each perturbation order.

You are right that these calculations have been done in a single (but arbitrary) reference frame. If we want to obtain the Hamiltonian in a moving frame, we would need to apply the boost generator to it. Due to the principle of relativity, the Hamiltonian in the moving frame will have the same expansion coefficients with respect to to a/c operators in the moving frame as the coefficients of the rest-frame Hamiltonian with respect to rest-frame a/c operators. So, if the rest-frame Hamiltonian is separable, then the moving-frame Hamiltonian is separable too.

I am not sure where you found any evidence that I adhere to the preferred frame idea. I am 100% behind the principle of relativity. All inertial frames are equal. It is true that specific calculations can be done more easily in a specific frame. And that's what I do. But this should not be a problem, since there exist well-defined rules about how to translate our descriptions between different frames.

Eugene.

 

[meopemuk]

While searching for information, I came across the following unrelated remark on p. 426: ''We can record time even if we do not measure anything, even if there is no physical system to observe.'' I wonder how you record time without observing (i.e., measuring) a clock. If there is no physical system to observe, there is in particular no clock and no observer.

I've tried to explain my views on measurements (and on time measurements, in particular) on pages xxix and xxx that refer to Figure 1. In this figure there is a clear separation between the measuring device and the observed physical system. The measuring device and the clock are not parts of the physical system. They are parts of the experimental equipment, which every observer must have. The clock plays a specific role in a sense that it records something (time), which has no relevance to the observed physical system. Time is a quantity that exists by itself, without any connection to the observed system. And time can be recorded even if there is no system to observe, i.e., if the space between the preparation device and the measuring apparatus in Fig. 1 is empty. This allows me to say in subsection 11.3.4 that there cannot be a "time observable"

Of course, we may decide to treat our clock as a physical system, i.e., put it between the preparation device and the measuring apparatus in Fig. 1. Then we can find all kinds of quantum uncertainties associated with the clock. E.g., we can find that the leads cannot have certain velocities and positions simultaneously. But then, in order to keep time labels of our measurements, we would need to choose some other (reference) clock in our laboratory. The readings of this reference clock will be considered as true classical time labels.

So, any laboratory has a clock associated with it. The readings of this clock are postulated to be exact and classical time labels that are attached to all measurements that we do in the laboratory.

It might also happen that our reference clock behaves irregularly due to quantum fluctuations. This simply means that we've chosen a bad instrument to serve as a clock. We would need to replace it with some other (e.g., more massive) device whose ticks are more regular and predictable.

This is kind of philosophical point, and I am still struggling to formulate it in a coherent fashion.

And on p.427; ''the formally quantum nature of clocks and rulers does not play any role in experimental physics.'' But they play a big role if the times and distances become short enough. Accurate experiments become impossible (or extremely difficult) when these are so short that the quantum nature starts playing a role.

I agree that this is not a well-formulated sentence. I am going to replace it with the following: "So, for theoretical purposes, it is reasonable to assume the availability of ideal clocks and rulers, whose performance is not affected by quantum effects."

Eugene.

 

[A. Neumaier]

If you want to make fun of Kita's paper, I can help you. One of these 7 citations is in my book, two of them are by my friend Shirokov in Dubna, and three are Kita's self-citations. So, we are not talking about mainstream here.

I had no intention to be funny here. ''useful'' is a sociological term with a different meaning of ''mainstream''. Mainstream is the dominant view on a subject; useful is something if people start using it once its exist.

I wouldn't regard Polyzou's work on covariant few-particle systems as mainstream. But it has about 200 citations, and looking at some of them, one finds that is useful to some extent. On the other hand, something that hasn't been used in many years can hardly be called useful.

 

[A. Neumaier]

I agree that the interacting field built in this manner is useless. It is useless not because it is non-covariant or non-commuting, but because actual calculations of the S-matrix, bound states, time evolution, etc. do not involve this field at all.

As can be seen by Weinberg, who odes scattering calculations via LSZ and bound state calculations via Bethe-Salpeter, these calculations instead involve the usual covariant fields. So do time evolution calculations in the CTP framework.

All we need for practical calculations is the Hamiltonian, and there is no need to worry about the properties of the interacting field.

You calculations with explicit Hamiltonians are less practical than those employed by Weinberg and CPT. As with ''useful'', ''practical'' means that it is reasonably easy to practice. This is not the case with your complex formulas.

In building the "dressed particle" version of QED I start from the usual QED Hamiltonian, [...] obtain the dressed particle interaction by applying an unitary transformation $$e^{i\Phi}$$ to the QED Hamiltonian.

This transformation is not unitary; see the remark at the bottom of p.4 of Shirokov's paper at http://lanl.arxiv.org/abs/math-ph/0703021 , that you quoted in one of your recent mails. Indeed, the paper is very nice in that it makes explicit much of what we had been talking about without introducing unnecessary details.

In the terminology of Shirokov's paper (but in my words), Haag's theorem is essentially saying that if you have a dressing transform W of the kind described in the paper then W is not unitarily implementable since the physical Hilbert space (which is the image of the dressed Fock space under the mapping W) is not a Fock space.

Note that _all_ rigorous constructions in constructive field theory that we had been talking about (though not the Wightman approach, which is considered ''axiomatic'', not ''constructive'') are precisely about the rigorous construction of a (nonunitary) dressing transformations that provides a definition of the physical Hilbert space on which the Heisenberg fields act.

The mistake in your experimental interpretation of your formalism is that you take as the physical Hilbert space the Fock space rather than its image under W. This gives all your ''physical'' operators a touch of surrealism revealed by the strange deviations from standard relativity reported e.g. in your Section 11.4. (E.g., at the bottom of p.437, you talk about a non-covariant boost transformation law - a contradiction in itself under the usual interpretation.)

As I claim in Theorem 10.2, the Hermitian operator $$\Phi$$ must be smooth (=separable) in order to preserve the S-matrix.

More importantly, it must be self-adjoint, which is not the case because of Haag's theorem (in the form mentioned above).

 

[meopemuk]

As can be seen by Weinberg, who odes scattering calculations via LSZ and bound state calculations via Bethe-Salpeter, these calculations instead involve the usual covariant fields. So do time evolution calculations in the CTP framework.

As we agreed already, my approach is not a "field theory". So, I don't use fields in any of those calculations. I use the old-fashioned quantum mechanics, which is based on the Hamiltonian.

This transformation is not unitary; see the remark at the bottom of p.4 of Shirokov's paper at http://lanl.arxiv.org/abs/math-ph/0703021 , that you quoted in one of your recent mails. Indeed, the paper is very nice in that it makes explicit much of what we had been talking about without introducing unnecessary details.

First, my approach is different from the one used by Shirokov. The difference is explained in subsection 10.2.10. I apply the dressing transformation to the Hamiltonian (and other Poincare generators). Particle states are not transformed, so the Fock space structure remains the same as in the free theory. Second, if the dressing operator $$e^{i \Phi}$$ were non-unitary I would get a non-Hermitian dressed Hamiltonian as a result. However, this Hamiltonian is explicitly represented as a sum of commutators of Hermitian operators, so it is bound to be Hermitian itself. See eq. (10.8). It might happen that this infinite series of Hermitian terms converges to something non-Hermitian in the infinite perturbation order limit. I am not sure about that. But I am not claiming that I have a non-perturbative approach. So, for my purposes it is safe to say that the dressed Hamiltonian is Hermitian and the dressing transformation is unitary.

In the terminology of Shirokov's paper (but in my words), Haag's theorem is essentially saying that if you have a dressing transform W of the kind described in the paper then W is not unitarily implementable since the physical Hilbert space (which is the image of the dressed Fock space under the mapping W) is not a Fock space.

I have a different understanding of what this paper says. It is best captured by the abstract itself:

It is demonstrated that the ”dressed particle” approach to relativistic local
quantum field theories does not contradict Haag’s theorem. On the contrary,
”dressing” is the way to overcome the difficulties revealed by Haag’s theorem."

The mistake in your experimental interpretation of your formalism is that you take as the physical Hilbert space the Fock space rather than its image under W. This gives all your ''physical'' operators a touch of surrealism revealed by the strange deviations from standard relativity reported e.g. in your Section 11.4. (E.g., at the bottom of p.437, you talk about a non-covariant boost transformation law - a contradiction in itself under the usual interpretation.)

As I said above, in my approach the Fock space is *not* transformed by W. You are right that the non-covariant boost transformation law is one of the most important conclusions of my work. If you found a specific mistake in the derivation of this law, I would appreciate your pointing it to me. In subsection 11.3.1 I claim that existing derivations of Lorentz transformations in special relativity are not correct, because they disregard interactions between particles. If you see a hole in my arguments, please let me know.

Eugene.

 

[A. Neumaier]

Particle states are not transformed, so the Fock space structure remains the same as in the free theory. Second, if the dressing operator $$e^{i \Phi}$$ were non-unitary I would get a non-Hermitian dressed Hamiltonian as a result.

The first statement is the same as what Shirokov does. Therefore, the second statement is true in a sharpened form: By (Haag's theorem or) Shirokov's remark (p.4 bottom)

the expression exp(R(a^*a) is not an operator, as it fails to map vectors of the Hilbert space in the Fock representation of operator alpha to vectors in the same space

your e^{iPhi} is undefined (except as a formal power series). Thus your dressed Hamiltonian is also undefined (except as a formal power series), and any claims about preserving properties based on the unitarity od e^{iPhi} are vacuous.

As I said above, in my approach the Fock space is *not* transformed by W.

Precisely, and this is the reason why you work in an unphysical representation that loses both the covariant meaning of the field coordinates and all field information. It is the source of all the strange things you report in Chapter 11.

You are right that the non-covariant boost transformation law is one of the most important conclusions of my work. If you found a specific mistake in the derivation of this law, I would appreciate your pointing it to me. In subsection 11.3.1 I claim that existing derivations of Lorentz transformations in special relativity are not correct, because they disregard interactions between particles.

They don't disregard anything since they stick to the physical representation (on the image of Fock space under W) rather than insisting (without sufficient reasons) that Fock space is physical.Your most important conclusion is based on this false identification of the physics.

Note that relativity was discovered through the Maxwell equations, and until after QED was first conceived, it was (apart from the single other observed fact - the mercury anomalies) the only reason to look for a marriage between quantum mechanics and relativity. The arguments of the maxwell fields have the traditional physical interpretation essential to give relativity a meaning. And the meaning of the arguments is preserved in the standard formulation of QED, while it is mutilated in your approach.

 

[meopemuk]

your e^{iPhi} is undefined (except as a formal power series). Thus your dressed Hamiltonian is also undefined (except as a formal power series), and any claims about preserving properties based on the unitarity od e^{iPhi} are vacuous.

I have stated repeatedly that I don't claim a non-perturbative solution. Everything I do is within perturbation theory, and only few lowest orders have been explored explicitly.

Precisely, and this is the reason why you work in an unphysical representation that loses both the covariant meaning of the field coordinates and all field information. It is the source of all the strange things you report in Chapter 11.

They don't disregard anything since they stick to the physical representation (on the image of Fock space under W) rather than insisting (without sufficient reasons) that Fock space is physical.Your most important conclusion is based on this false identification of the physics.

Discussion in Section 11.2, where I derive the non-covariance of boost transformations, is completely unrelated to the Fock space or dressing transformations or other subtle matters. This discussion even doesn't require quantum mechanics, because all arguments can be repeated for classical particles moving along trajectories. The only essential thing is that there exists an interacting representation of the Poincare group (in the Hilbert space or in the phase space). In this representation the generator of boosts is interaction-dependent (see Weinberg's (3.3.20)). From this it follows immediately that boost transformations cannot have the same universal covariant form as in the non-interacting case.

Eugene.

 

[A. Neumaier]

I have stated repeatedly that I don't claim a non-perturbative solution. Everything I do is within perturbation theory, and only few lowest orders have been explored explicitly.

Then why bother with Haag's theorem at all? It says nothing in perturbation theory. Also the distinction between Fock representations and non-Fock representations does not even exist in perturbation theory. So you should stop claiming anything about that all states are representable in Fock space! They are so only in perturbation theory.

But perturbation theory is known to be very inadequate for particle physics - with exception of the processes where all external lines are massive elementary particles. QED is not among these theories, since the photon is massless. Even there, you need some nonperturbative tricks such as a resummation of propagators, in order to get the correct form factors and self-energies (which are apparently missing in your perturbative treatment).

Thus if you want to claim to have a theory of QED, you need to go beyond perturbation theory and beyond Fock space!

Discussion in Section 11.2, where I derive the non-covariance of boost transformations, is completely unrelated to the Fock space or dressing transformations or other subtle matters. This discussion even doesn't require quantum mechanics, because all arguments can be repeated for classical particles moving along trajectories.

I didn't spent enough time on the details of your theory to figure out where exactly is the mistake. (I am not interested in understanding the details of a theory whose conclusions violate so obviously the demands of relativity.)

The no-go theorem by Currie, Jordan, and Sudarshan proves that the multi-particle view for classical point particles is inconsistent with relativity. You comment this on p.418 with ''Of course, it is absurd to think that there are no interactions in nature.'' But it only proves the absence of point particles, not the absence of interactions between fields. Your conclusion on p.419, ''for us the only way out of the paradox is to admit that Lorentz transformations of special relativity are not applicable to observables of interacting particles'' may satisfy you, but it doesn't satisfy anyone who understood the meaning of covariance. A field theory has no need to make such queer assumptions.

The only essential thing is that there exists an interacting representation of the Poincare group (in the Hilbert space or in the phase space).

And your existence claim for this representation in the quantum case is solely based on the unitarity of the operator e^{iPhi}, that, according to Shirokov (or Haag) doesn't exist.

 

[A. Neumaier]

Note that relativity was discovered through the Maxwell equations, and until after QED was first conceived, it was (apart from the single other observed fact - the mercury anomalies) the only reason to look for a marriage between quantum mechanics and relativity. The arguments of the Maxwell fields have the traditional physical interpretation essential to give relativity a meaning.

See post #27 in https://www.physicsforums.com/showthread.php?t=474571

 

[meopemuk]

Thus if you want to claim to have a theory of QED, you need to go beyond perturbation theory and beyond Fock space!

First, we agreed that my approach is *not* a field theory. The entire Part II of the book has the title "The quantum theory of particles". So, it is not the same as QED. The claim is that within this particle-based formalism it is possible to reproduce the same experimental observations as in QED and even more (e.g., the time evolution). I understand that this claim sounds hollow, because I haven't presented a single loop calculation within my approach. This is related to infrared difficulties, as we discussed earlier.

Second, I am not going to back away from the Fock space, because this structure of the Hilbert space (with orthogonal n-particle sectors etc.) is postulated in my approach. I don't see any fundamental problem in representing infinite number of soft photons in the Fock space. This is not a bigger problem than placing the infinity on a real line. We've talked about that already.

(I am not interested in understanding the details of a theory whose conclusions violate so obviously the demands of relativity.)

I am pretty sure that my conclusions do not violate the "principle of relativity", i.e., the equality of all inertial frames. I am also sure that my conclusions do violate conclusions of Einstein's special relativity, such as the covariance, Minkowski space-time, etc. However, there is no logical contradiction, because covariance and Minkowski space-time were derived in special relativity by using a tacit assumption of the absence of interactions. I am working in the interacting case. I go through all these arguments in detail in Chapter 11.

And your existence claim for this representation in the quantum case is solely based on the unitarity of the operator e^{iPhi}, that, according to Shirokov (or Haag) doesn't exist.

I would like to steer our discussion away from things like the difference between Hermitian and self-adjoint operators. The existence of the unitary dressing transformation e^{iPhi} is not the only foundation of my approach. Actually, this transformation serves only as a demonstration of the connection betwen the old theory (QED) and the new approach. But this new approach can stand on its own as well. For its viability the only important thing is the existence of 10 Poincare generators with usual commutators. These commutators have been proven in Appendix O. And this proof is completely unrelated to the issue of self-adjointness of Phi.

Eugene.

 

[A. Neumaier]

First, we agreed that my approach is *not* a field theory. The entire Part II of the book has the title "The quantum theory of particles". So, it is not the same as QED.

It is even designed as a theory to _replace_ field theory,: On p.353 you write:''The position taken in this book is that the presence of infinite counterterms in the Hamiltonian of QED H^c is not acceptable and that the Tomonaga-Schwinger-Feynman renormalization program was just a first step in the process of elimination of infinities from quantum field theory. In this section we are going to propose how to make a second step in this direction: remove infinite contributions from the Hamiltonian H^c and solve the paradox of ultraviolet divergences in QED.'' This is a claim that you have something _better_ than QED, not just something _different_.

If it should not be a replacement for QED, why do you argue so heavily against QED? But you beat a straw man only:

You begin it with Section ''10.1 Troubles with renormalized QED'', although QED has none of the trouble you have to fight with. On p.349 you introduce a straw man with the words ''The traditional interpretation of the renormalization approach is that infinities in the Hamiltonian H^c (9.38) have a real physical meaning. The common view is that bare electrons and protons really have infinite masses ˜m and ˜M and infinite charges +-~e''. But this is far from the traditional interpretation, which (in the more careful treatments) declares all bare stuff to be devoid of meaning, and the infinities to be a limit that is to be taken only at the end of the renormalization calculations. Moreover, working with a cutoff Lambda and taking m,M,e finite but large gives formulas that are essentially as accurate as the infinite limits, and certainly as accurate as you low order approximations.

Another strawman is introduced on p.350: ''traditional QED predicts rather complex dynamics of the vacuum and one-particle states.'' This is not the case; the vacuum is stable under the renormalized dynamics, and the space of all 1-particle states is invariant under the dynamics in the (formal) Wightman representation. You get things wrong since you consider the dynamics of unphysical bare vacuum and 1-particle states, which no quantum field theorist considers.

Your complaints about the finite-time behavior of QED on p.352 are unfounded, as we have discussed already. Lots of applications of QED on the kinetic or hydrodynamic level are time-dependent, and consistent with experiments.

On p.383, you write: ''In spite of its dominant presence in theoretical physics, the true meaning of QFT and its mathematical foundations are poorly understood.'' This is not ttrue; except if you talk about your personal understanding (but then you should say so). The mathematics of perturbative QED is perfectly understood, in full mathematical rigor (see Steinmann's book ''Perturbative quantum electrodynamics and axiomatic field theory''). What you offer as replacement is much is less rigorous. As to the true meaning - this is a subjective term -- if you think your noncovariant conclusions are in any reasonable sense truer than the covariant conclusions of QED, you are mistaken.

On p.384, you write ''If (as usually suggested) fields are important ingredients of physical reality, then we should be able to measure them. However, the things that are measured in physical experiments are intimately related to particles and their properties, not to fields. For example, we can measure (expectation values of) positions, momenta, velocities, angular momenta, and energies of particles as functions of time (= trajectories). [...] All these measurements have a transparent and natural description in the language of particles and operators of their observables.'' But you ignore that the routine measurements of the macroscopic electromagnetic field are measurements of (expectation values of) the fields E(x) and B(x), on precisely the same par as the observables you list. None of these measurements has a transparent and natural description in the language of particles and operators of their observables.

You call the latter ''very questionable. When we say that we have “measured the electric field” at a certain point in space, we have actually placed a test charge at that point and measured the force exerted on this charge by surrounding charges.'' But when we measure the momentum of a particle in a scattering experiment, we have actually placed a wire chamber in its way, together with a magnetic field, and measured the energy deposited on the wires, from which we calculated the curvature of the track and deduced the momentum at the time the particle entered the chamber. - And you forget to question this indirect measurement. Measurement of angular momentum or the energy of bound states are also measured quite indirectly, and hence should be very questionable according to your standards!

Thus - a double standard wherever one looks. You praise your achievements, silently glossing over their weaknesses, and you magnify the problems of QED to an extent where you can't hope to get agreement from anyone who understands the matter better than you do.

The claim is that within this particle-based formalism it is possible to reproduce the same experimental observations as in QED and even more (e.g., the time evolution). I understand that this claim sounds hollow, because I haven't presented a single loop calculation within my approach. This is related to infrared difficulties, as we discussed earlier.

It is hollow, and sounds so. No computation of form factors, or of self energies, or of the lamb shift, or of the anomalous magnetic moment of the electron. What you calculated is almost disjoint of what is calculated (and compared with experiment) in the usual books.

Note that in the standard treatment (e.g., Peskin & Schroeder (6.59)), the computation of the anomalous magnetic moment to lowest nontrivial order is free of infrared problems.
If you run into infrared problems there, these are introduced by your problematic dressing transform.

Second, I am not going to back away from the Fock space, because this structure of the Hilbert space (with orthogonal n-particle sectors etc.) is postulated in my approach. I don't see any fundamental problem in representing infinite number of soft photons in the Fock space. This is not a bigger problem than placing the infinity on a real line. We've talked about that already.

Talked yes, but not substantiated.

I would like to steer our discussion away from things like the difference between Hermitian and self-adjoint operators. The existence of the unitary dressing transformation e^{iPhi} is not the only foundation of my approach. Actually, this transformation serves only as a demonstration of the connection between the old theory (QED) and the new approach.

No. It is the _basis_ of your whole approach. Without the e^{iPhi} conjugation, you don't even have a Hamiltonian to start with. You borrow the theory and the S-matrix from standard QED, and then you transform the nice, covariant results of the latter into a mutilated version that has all sorts of noncovariant features.

 

[meopemuk]

You begin it with Section ''10.1 Troubles with renormalized QED'', although QED has none of the trouble you have to fight with. On p.349 you introduce a straw man with the words ''The traditional interpretation of the renormalization approach is that infinities in the Hamiltonian H^c (9.38) have a real physical meaning. The common view is that bare electrons and protons really have infinite masses ˜m and ˜M and infinite charges +-~e''. But this is far from the traditional interpretation, which (in the more careful treatments) declares all bare stuff to be devoid of meaning, and the infinities to be a limit that is to be taken only at the end of the renormalization calculations. Moreover, working with a cutoff Lambda and taking m,M,e finite but large gives formulas that are essentially as accurate as the infinite limits, and certainly as accurate as you low order approximations.

By referring to calculations with finite cutoff Lambda and very large m,M,e you basically confirm my statement that standard approaches use the idea of infinite (or very large, which is basically the same) particle parameters. Just after this quote I go on to explain the usual idea of physical particles as linear combinations of bare particle states. Perhaps you are right that different textbooks use different philosophies on whether bare and virtual particles should be regarded as something real or imaginary. So, I should avoid words like "traditional interpretation" and "common view". I will replace them with "In some textbooks", "sometimes", etc.

Another strawman is introduced on p.350: ''traditional QED predicts rather complex dynamics of the vacuum and one-particle states.'' This is not the case; the vacuum is stable under the renormalized dynamics, and the space of all 1-particle states is invariant under the dynamics in the (formal) Wightman representation. You get things wrong since you consider the dynamics of unphysical bare vacuum and 1-particle states, which no quantum field theorist considers.

Thanks, I will replace this phrase with ''traditional QED predicts rather complex dynamics of *bare* vacuum and one-particle states.'' Yes, I agree that dynamics of bare states is useless. But, unfortunately, the Hamiltonian of QED is formulated explicitly in terms of bare particle operators only. So, it is not possible to describe the time dynamics of physical particles without special tricks. Moreover, when quantum field theorists calculate scattering amplitudes they are pretty happy to identify states created by bare creation operators as real observable particles. Perhaps, axiomatic QFT can deal with these problems nicely, but these things are not explained in textbooks, like Weinberg.

Your complaints about the finite-time behavior of QED on p.352 are unfounded, as we have discussed already. Lots of applications of QED on the kinetic or hydrodynamic level are time-dependent, and consistent with experiments.

I am not convinced about that. If dynamical solution does not exist even for the simplest 2-particle interacting system, then how one can be sure about the correctness on the more complicated kinetic or hydrodynamic level?

Ideally, I would like to see a time-dependent wave function for a system of two slowly colliding particles obtained in QED from first principles. This solution should satisfy unitarity, agree with simple QM and classical solutions, yield the same scattering probabilities as the S-matrix approach. There will be approximations, for sure, but they must be clearly justified. Then I will be convinced that QED can describe the time evolution.

On p.383, you write: ''In spite of its dominant presence in theoretical physics, the true meaning of QFT and its mathematical foundations are poorly understood.'' This is not ttrue; except if you talk about your personal understanding (but then you should say so). The mathematics of perturbative QED is perfectly understood, in full mathematical rigor (see Steinmann's book ''Perturbative quantum electrodynamics and axiomatic field theory''). What you offer as replacement is much is less rigorous. As to the true meaning - this is a subjective term -- if you think your noncovariant conclusions are in any reasonable sense truer than the covariant conclusions of QED, you are mistaken.

Thanks. I will replace "are poorly understood" with "remain controversial". The quotes from Wilczek and Wallace ephasize this controversy.

On p.384, you write ''If (as usually suggested) fields are important ingredients of physical reality, then we should be able to measure them. However, the things that are measured in physical experiments are intimately related to particles and their properties, not to fields. For example, we can measure (expectation values of) positions, momenta, velocities, angular momenta, and energies of particles as functions of time (= trajectories). [...] All these measurements have a transparent and natural description in the language of particles and operators of their observables.'' But you ignore that the routine measurements of the macroscopic electromagnetic field are measurements of (expectation values of) the fields E(x) and B(x), on precisely the same par as the observables you list. None of these measurements has a transparent and natural description in the language of particles and operators of their observables.



You call the latter ''very questionable. When we say that we have “measured the electric field” at a certain point in space, we have actually placed a test charge at that point and measured the force exerted on this charge by surrounding charges.'' But when we measure the momentum of a particle in a scattering experiment, we have actually placed a wire chamber in its way, together with a magnetic field, and measured the energy deposited on the wires, from which we calculated the curvature of the track and deduced the momentum at the time the particle entered the chamber. - And you forget to question this indirect measurement. Measurement of angular momentum or the energy of bound states are also measured quite indirectly, and hence should be very questionable according to your standards!

This is rather philosophical dispute, which we are not going to resolve easily. As you said, the war between particles and fields goes on for centuries. Of course, many measurements are done indirectly. My point is that when we happen to measure something in the most direct way, like the photon blackening a grain of photoemulsion, we always see countable indivisible particles.

Thus - a double standard wherever one looks. You praise your achievements, silently glossing over their weaknesses, and you magnify the problems of QED to an extent where you can't hope to get agreement from anyone who understands the matter better than you do.

Hopefully, with your help I'll make the presentation in the book less abrasive.

Note that in the standard treatment (e.g., Peskin & Schroeder (6.59)), the computation of the anomalous magnetic moment to lowest nontrivial order is free of infrared problems.
If you run into infrared problems there, these are introduced by your problematic dressing transform.

This is actually a good idea! Why didn't I think about it before? I can calculate this part of the charge-charge dressed potential separately since no infrared infinities should be involved.
Thank you very much.


Eugene.

 

[A. Neumaier]

Thanks, I will replace this phrase with ''traditional QED predicts rather complex dynamics of *bare* vacuum and one-particle states.''

But this is not true either. QED predicts nothing at all about bare states, since these disappear completely during renormalization. In the limit where the cutoff is removed, the bare stuff no longer exists in any meaningful sense!

The predictions of QED (or any other QFT) solely concern expectation values and time-ordered expectation values of products of the physical (=renormalized) Heisenberg fields. This (and only this) is what contains the physics, and this (and only this) can be compared with experiment!

Yes, I agree that dynamics of bare states is useless. But, unfortunately, the Hamiltonian of QED is formulated explicitly in terms of bare particle operators only.
No. This is only the starting point, not the final Hamiltonian.

Your theory begins with the same starting point and then performs nonexisting ''unitary transformations'' in order to translate the bare, ill-defined stuff into something perturbatively well-defined (ignoring infrared problems, which prove the lack of self-adjointness of your Fock space Hamiltonian). You then regard _this_ as the ''real'' theory - the other stuff was just scaffolding to be thrown away after you have the Hamiltonian.

In the same way, standard QED begins with the same starting point and then performs renormalization calculations in order to translate the bare, ill-defined stuff into something perturbatively well-defined. _This_ is then regarded as the ''real'' theory - the other stuff was just scaffolding to be thrown away after you have the Hamiltonian.

What you accept as only a pretext for your theory should also be treated by you as only a pretext for standard QED. This is how the bare stuff is viewed by the experts, and to be fair, you should view it in the same way.

So, it is not possible to describe the time dynamics of physical particles without special tricks.

The tricks are no worse than applying your nonexisting operator exp{iPhi} to get what you take as your basis.

Moreover, when quantum field theorists calculate scattering amplitudes they are pretty happy to identify states created by bare creation operators as real observable particles.

_Nobody_ is doing that.

Perhaps, axiomatic QFT can deal with these problems nicely, but these things are not explained in textbooks, like Weinberg.

How wrong you are! Maybe you haven't noticed that because of your aversion to filed theoretic methods (which you denounce as mere mathematical tricks), but it is in every textbook where the LSZ formula is derived. For example, in Weinberg, this is handled in Section 10.2.

Note that on p.430, Omega_0 is the true vacuum, not the bare one, and the A_i are the renormalized Heisenberg fields with space-time arguments. The latter act on the physical Hilbert space spanned by the states of the form A_1 ... A_n Omega_0, though Weinberg doesn't emphasize this explicitly. But one can see it from the fact that he takes matrix elements between such states (_not_ between bare states!). This is _precisely_ the recipe that I had given in my explanation of the Wightman approach to QFT. Wightman didn't take his approach from nowhere, but only isolated the minimal stuff from the usual, nonrigorous treatment that one would have to make clear mathematical sense of in order to have a rigorous, nonperturbatively defined theory.

I am not convinced about that. If dynamical solution does not exist even for the simplest 2-particle interacting system, then how one can be sure about the correctness on the more complicated kinetic or hydrodynamic level?

Dynamical solutions exist for the whole physical Hilbert space, and the field operators from which the kinetic and hydrodynamic equations are derived act on this space! Iit is just that the notion of a 2-particle system is no longer well-defined, except asymptotically.

Ideally, I would like to see a time-dependent wave function for a system of two slowly colliding particles obtained in QED from first principles.

It is you who is proposing a dynamical particle view of QED; so it is your obligation to substantiate that picture. Thus you have to study standard QED well enough that you can make a reasonable proposal for what in this standard framework the state of two slowly colliding particles should be. Then you get its evolution for free. The mainstream view is that the dynamics is a dynamics of fields, and particles exist only as asymptotic bound states. This is the version in which QED makes sense. Particles at finite times only exist in an approximate sense.

This solution should satisfy unitarity, agree with simple QM and classical solutions, yield the same scattering probabilities as the S-matrix approach. There will be approximations, for sure, but they must be clearly justified. Then I will be convinced that QED can describe the time evolution.

I gave you the construction of the Hilbert space and the spanning sets between which one computes the S-matrix elements. You can read Haag-Ruelle theory to find out how in simple cases (which apply to massive QED) the asymptotic particle states are constructed, and then guess from this the form of the approximation you need to make to get what you want.

Thanks. I will replace "are poorly understood" with "remain controversial". The quotes from Wilczek and Wallace emphasize this controversy.

This is foul play. If you criticize QED because it has no mathematically rigorous formulation so far, you must criticize your own theory for the same reason. Since you excuse your own theory from this demand, you have no moral right to call the understanding of QED controversial. The controversies are _only_ about the question how rigorous QED can be made. But _nothing_ about the experimental content of the theory is controversial!

This is rather philosophical dispute, which we are not going to resolve easily. As you said, the war between particles and fields goes on for centuries. Of course, many measurements are done indirectly.

You argue that momenta or angular momenta (where your theory happens to have observables) be truly observable, while electromagnetic fields (for which your theory has no observables) to be very questionably observable, though both require about the same degree of indirectness. What you actually write (since you hide the momentum indirection) is a very unfair and biased argument that no one will buy who has only a moderately realistic view of how actual measurement must be done.

My point is that when we happen to measure something in the most direct way, like the photon blackening a grain of photoemulsion, we always see countable indivisible particles.

So the position of photons is measurable in the most direct way, but you don't even have an observable for it in your theory! This shows that the observables in a theory and the naive intuition about measurements diverge quite radically! And as discussed in the other thread, you can never measure a photon while it is alive! This again shows the same thing!

Hopefully, with your help I'll make the presentation in the book less abrasive.

You'll not be able to hide the wolf (a faulty interpretation of QED and a faulty view of covariance) in sheep's clothing (aka less abrasive presentation). The second part of your book needs important corrections in the contents, not only in the presentation!

This is actually a good idea! Why didn't I think about it before? I can calculate this part of the charge-charge dressed potential separately since no infrared infinities should be involved.

If you like this sort of advice, I have two pieces more:

1. The photon self-energy is infrared finite to first nontrivial order; see Weinberg (11.2.16) and (11.2.22).

2. Why don't you postulate that the photon has a tiny mass? This is experimentally indistinguishable from real QED, and has a number of advantages:
-- Massive photons have a position operator, and hence a fully adequate Schroedinger picture. This would make your philosophical position much better grounded.
-- Massive photons save you from all infrared problems. Without the infrared problems, Fock space is perturbatively fully adequate, and all my criticism regarding the IR problem and wrong asymptotics is no longer applicable.
-- With massive photons, you can calculate radiative corrections to Compton scattering and get a finite result for the Lamb shift.

 

[meopemuk]

How wrong you are! Maybe you haven't noticed that because of your aversion to filed theoretic methods (which you denounce as mere mathematical tricks), but it is in every textbook where the LSZ formula is derived. For example, in Weinberg, this is handled in Section 10.2.

Note that on p.430, Omega_0 is the true vacuum, not the bare one, and the A_i are the renormalized Heisenberg fields with space-time arguments. The latter act on the physical Hilbert space spanned by the states of the form A_1 ... A_n Omega_0, though Weinberg doesn't emphasize this explicitly. But one can see it from the fact that he takes matrix elements between such states (_not_ between bare states!). This is _precisely_ the recipe that I had given in my explanation of the Wightman approach to QFT. Wightman didn't take his approach from nowhere, but only isolated the minimal stuff from the usual, nonrigorous treatment that one would have to make clear mathematical sense of in order to have a rigorous, nonperturbatively defined theory.

As far as I can tell, physical states and fields A_i are used only in abstract proofs, like in Weinberg's Chapter 10. This is because such objects have not been formulated in explicit forms necessary for real calculations. In actual QED calculations (see section 8.7 and Chapter 11) one still uses the bare particle picture, bare a/c operators and free fields.

This is foul play. If you criticize QED because it has no mathematically rigorous formulation so far, you must criticize your own theory for the same reason.

I am not criticizing QED for the lack of mathematical rigor. I am criticizing it, because it cannot offer a plausible solution for the time dynamics of the simplest 2-particle state. You said that in QED particles make sense in asymptotic states. Fine. Then let us prepare an asymptotic 2-particle state and let us follow its time evolution and see how its wave function changes in time. I guess that QED cannot do that. QED simply does not have an adequate Hamiltonian to do this job. This is my concern.

If you like this sort of advice, I have two pieces more:

1. The photon self-energy is infrared finite to first nontrivial order; see Weinberg (11.2.16) and (11.2.22).

This is a good one too! I guess that the corresponding dressed particle potential will be the same as Uehling potential in (11.2.38) and it will contribute a little bit to the Lamb shift (11.2.42)

2. Why don't you postulate that the photon has a tiny mass? This is experimentally indistinguishable from real QED, and has a number of advantages:
-- Massive photons have a position operator, and hence a fully adequate Schroedinger picture. This would make your philosophical position much better grounded.
-- Massive photons save you from all infrared problems. Without the infrared problems, Fock space is perturbatively fully adequate, and all my criticism regarding the IR problem and wrong asymptotics is no longer applicable.
-- With massive photons, you can calculate radiative corrections to Compton scattering and get a finite result for the Lamb shift.

I will think about this advice too. Thanks.

Eugene.

 

[A. Neumaier]

As far as I can tell, physical states and fields A_i are used only in abstract proofs, like in Weinberg's Chapter 10.

You can't tell far. All renormalized S-matrix calculations are done using the LSZ-formulas derived in Chapter 10 from this characterization of the physical states. Why else would Weinberg introduce the discussion of LSZ on p.429 by saying: ''will help us later in dealing with radiative corrections"? It is needed to define the renormalization recipes in a way that ensures correct scattering results between physical states!

From the LSZ formula (10.2.3), which is valid nonperturbatively (hence independent of any free fields), Weinberg gets on p.437-438 the usual Feynman rules (and he says so explicitly in the second sentence of p.438); the free field (or rather its propagator) appears at the bottom of p.437. But, as he emphasizes, for arbitrary spin, including bound states, and including the field renormalization factor 1/N (which was missing in the heuristic derivation in Chapter 6, Thereby the heuristic derivation is justified and improved. The notion of a bare versus renormalized field is introduced by Weinberg only on p.438/9, when he exploits LSZ to make the connection!

Weinberg urges the reader on p.292 that only the canonical approach gives reliable results in more complex situations, because the earlier heuristic derivation is no longer justified when there are rearrangement collisions, or when extra contact terms are needed in the Hamiltonian (see the footnotes on p.110 and p.145), and that ''when we come to non-Abelian gauge theories, in Volume II, this extra convenience will become a necessity''.

Of course, once one has the LSZ formula (10.2.3), one only needs to evaluate the right hand side using perturbation theory, and the latter simply uses the Wick rules for the free fields derived in Chapter 6 for scalar/spinor fields and in Chapter 8 for QED. (But this is no worse than your use of free fields in working out how your Hamiltonian acts on a given state.)

This is because such objects have not been formulated in explicit forms necessary for real calculations. In actual QED calculations (see section 8.7 and Chapter 11) one still uses the bare particle picture, bare a/c operators and free fields.

In Section 8.7 he only calculates Compton scattering without radiative corrections, which could already be done in 1930. At this order of accuracy, N=1. But once one includes radiative corrections, one needs the renormalized version derived in Chapter 10.
This is the reason why the 1-loop calculations are done in Chapter 11, with LSZ already proved!

I am not criticizing QED for the lack of mathematical rigor. I am criticizing it, because it cannot offer a plausible solution for the time dynamics of the simplest 2-particle state. You said that in QED particles make sense in asymptotic states. Fine. Then let us prepare an asymptotic 2-particle state and let us follow its time evolution and see how its wave function changes in time. I guess that QED cannot do that. QED simply does not have an adequate Hamiltonian to do this job. This is my concern.

Of course one can do it.It is just that nobody has been interested to work it out since one can never experimentally follow the time evolution of a real electron or photon.
Experimentally you prepare it somewhere, and you detect it somewhere, and from the statistics you can calculate cross sections. That's all.

If you believe that you can test the time evolution experimentally and therefore need the formulas to do so, it is _your_ job to derive then. QED tells you how to do it in principle, but the details (like with any calculation) must be done by those interested in the results. The way to do it is this: Given an arbitrary state Psi in the Wightman representation, where all its creation fields are taken at time t=0, you can compute its inner product with an asymptotic Fock state Phi by taking the inner product with the physical state Phi(t) where all its creation fields are at the same time t and letting t --> -inf. Now pick Psi such that to the desired accuracy only the 2-particle contributions survive, and you have your desired 2-particle in state at time t=0. You can propagate it in time simply by replacing the time t=0 by an arbitrary time.

It is not necessary to do this explicitly in order to see that it can be done. Actually doing it is worth only for someone who believes that such fictions as exact 2-particle in states exist and have a useful explicit time evolution. I am not among those, and hence will not do it.

 

[meopemuk]

Of course one can do it.It is just that nobody has been interested to work it out since one can never experimentally follow the time evolution of a real electron or photon.
Experimentally you prepare it somewhere, and you detect it somewhere, and from the statistics you can calculate cross sections. That's all.

In the low energy macroscopic world of our everyday experience it is possible to see time-dependent trajectories of charged particles (charged drops of oil, or specks of dust or whatever) interacting with each other. So, a complete theory should be able to calculate not only scattering cross-sections, but also time-dependent evolution of wave functions.

The way to do it is this: Given an arbitrary state Psi in the Wightman representation, where all its creation fields are taken at time t=0, you can compute its inner product with an asymptotic Fock state Phi by taking the inner product with the physical state Phi(t) where all its creation fields are at the same time t and letting t --> -inf. Now pick Psi such that to the desired accuracy only the 2-particle contributions survive, and you have your desired 2-particle in state at time t=0. You can propagate it in time simply by replacing the time t=0 by an arbitrary time.

Why don't you just write the Hamiltonian? In quantum mechanics the Hamiltonian contains all one needs to know to describe the time evolution. Every relativistic quantum theory must have 10 Hermitian generators of the Poincare group, one of which is the Hamiltonian. If Wightman approach is such a theory, then it must have an explicitly written Hamiltonian.

If you are not comfortable of doing this in electrodynamics, then please write the Hamiltonian of a 2-dimensional Phi^4 theory, where everything is understood, as you say. If I understand correctly, the Wightman et al. approach permits creation of two separated physical bosons moving toward each other. I presume that at each later time point one can measure (the probabilities of) how many physical particles there are in the system and what are their observables. This time evolution must be unitary, so there should exist a Hermitian Hamiltonian responsible for it.

It is not necessary to do this explicitly in order to see that it can be done. Actually doing it is worth only for someone who believes that such fictions as exact 2-particle in states exist and have a useful explicit time evolution. I am not among those, and hence will not do it.

I have no problem with the fact that initial asymptotic 2-particle state may acquire components in n-particle sectors as time evolution progresses and the two original particles move closer to each other and interact stronger. The full Hamiltonian should be able to describe the appearance of such n-particle contributions. However, it is known from experiment that for slowly moving projectiles these n-particle contributions are negligible. For example, one usually neglects radiation processes in the theory of electronic circuits, where electrons move very slowly.

Eugene.

 

[A. Neumaier]

In the low energy macroscopic world of our everyday experience it is possible to see time-dependent trajectories of charged particles (charged drops of oil, or specks of dust or whatever) interacting with each other. So, a complete theory should be able to calculate not only scattering cross-sections, but also time-dependent evolution of wave functions.

But a charged drop of oil is macroscopic, and well described by the classical Maxwell equations, as every engineer can tell you. Thus one only needs to look at the macroscopic limit of QED - and I had already described how to get that.

Why don't you just write the Hamiltonian?

$$H \psi(x_1,t_1,...x_N,t_N) = -i\sum_k d/dt_k \psi(x_1,t_1,...x_N,t_N)$$
I wonder why you couldn't derive that yourself from the prescription I gave.
The explicit form adds nothing to the description.

I have no problem with the fact that initial asymptotic 2-particle state may acquire components in n-particle sectors as time evolution progresses and the two original particles move closer to each other and interact stronger. The full Hamiltonian should be able to describe the appearance of such n-particle contributions.

It does, in the way I described.

 

[meopemuk]

$$H \psi(x_1,t_1,...x_N,t_N) = -i\sum_k d/dt_k \psi(x_1,t_1,...x_N,t_N)$$
I wonder why you couldn't derive that yourself from the prescription I gave.
The explicit form adds nothing to the description.

Few questions:

1. Can I interpret \psi as a wave function? I.e., is the square of \psi the probability density?

2. What is the meaning of N? Is it the number of particles?

3. Why there are N time labels?

4. What is the t-dependence of \psi? Without such explicit t-dependence the Hamiltonian remains undefined.

5. How this form can be used to calculate the interacting time evolution of an initial 2-particle state?

Thanks.
Eugene.